Chapter 12.4, Problem 32PPS

a.

To determine

To find: whether the selection involves a permutation or a combination.

Answer to Problem 32PPS

The selection involves a permutation.

Explanation of Solution

Given information:

51 contestants are there in The Teen Miss USA pageant.

Teen Miss USA and 4 runners-up are chosen by the judges.

Here, we have to find that the selection involves a permutation or a combination.

Since the order is important, then this is a permutation problem.

Hence, the selection involves a permutation.

b.

To determine

To find: the number of ways that the judges can choose Teen Miss USA and 4 runners-up.

Answer to Problem 32PPS

The judges can choose Teen Miss USA and 4 runners-up in 281887200 ways.

Explanation of Solution

Given information:

51 contestants are there in The Teen Miss USA pageant.

Teen Miss USA and 4 runners-up are chosen by the judges.

Formula:

  $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

To calculate the number of ways that the judges can choose Teen Miss USA and 4 runners-up,

  $\begin{array}{l}P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}\\ P\left(51,5\right)=\frac{51!}{\left(51-5\right)!}\left[n=51\text{ and }r=5\right]\\ =\frac{51!}{46!}\\ =\frac{51\times 50\times 49\times 48\times 47\times 46!}{46!}\left[\text{Divide by common factor}\text{.}\right]\\ =281887200\end{array}$

Hence, the judges can choose Teen Miss USA and 4 runners-up in 281887200 ways.